Enhancing convergence efficiency of self-propelled agents using direction preference

Highlights

• A direction preference model is proposed for self-propelled agent systems.

• Agents prefer to synchronize with their neighbors moving in a certain direction, which is called preference direction.

• There exists a universal optimal preference direction in noise-free environment.

• The optimal direction preference model can accelerate the synchronization speed in noise-free environment.

Abstract

In this paper, we investigate the effect of direction preference on self-propelled agents. In the well-known Vicsek model, an agent treats its neighbors equally and updates its direction by the average direction of its neighbors. Whereas, in this study, agents prefer to synchronize with their neighbors moving in a certain direction, which is called preference direction. The center agent judges the influence value from its neighbor according to the direction angle between them. We assume that there exists a preference direction angle $\beta$. When the direction angle between the neighbor and the center agent is closer to $\beta$, the neighbor has greater influence on the center agent. The modified Vicsek model with preference direction angle is called direction preference model. We use the parameter $\alpha$ $(0<\alpha \le 1)$ to adjust the effect of direction preference. The larger the value of $\alpha$, the weaker the effect of direction preference. If $\alpha =1$, the direction preference will lose its effect. In the simulation experiments, different values of $\beta$ and $\alpha$ are discussed. Simulation results demonstrate that in noise-free environment the direction preference model with optimal $\beta =3\pi /8$ can accelerate the synchronization speed; in noise environment the direction preference model with $\beta$ be in $[3\pi /8,5\pi /8]$ has stronger robustness than the original Vicsek model and the optimal value of $\beta$ is altered.

Introduction

Collective behavior is very common in nature, such as bird flocks [1], [2], fish schools [3], [4], [5], [6], insect swarms [7], [8] and bacterial colonies [9], [10]. This amazing natural phenomenon has attracted many physicists, biologists and control scientists to study. They explore the nature of collective behavior through observation, simulation, and mathematical proof [11].

In 1987, Reynolds simulated the first collective motion through the computer program [12]. In his model, three rules of collective motion were proposed: aggregation, synchronization and separation. These three principles provide direction for the research of collective motion [13], [14], [15], [16], [17], [18], [19], [20], [21]. In terms of cluster synchronization, the most famous model was proposed by Vicsek [19], now well known as the Vicsek model (VM). In the VM, all agents are distributed in a periodic square environment randomly and have the same fixed speed and communication radius. In direction updating, each agent takes the average direction of all its neighbors (include itself) within the communication radius as its next step direction. The mechanism of the VM is very simple, but it can well simulate the phase transition process from disorder to order in nature. Because it is simple and efficient, the VM is considered as the basic model for collective motion. The VM has been further studied by other researchers [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. For instance, Ihle uses the quantitative kinetic theory to outline the used theory of the VM [24]. Romensky et al. believe that agents prefer to synchronize with their neighbors who are in a similar direction and focus their research on the effect of velocity restriction on the phase transition of the VM [34]. Piwowarczyk et al. study the influence of sensorial delay on clustering and swarming and find that short delays can enhance the emergence of clusters and swarms in VM [23]. Some researchers add the view angle to the VM and study the effect of view angle on collective motion [25], [26], [27], [28]. Li et al. modify the constant-speed VM into a variable-speed model, the studies show that the modified model is easier to achieve direction synchronization than the original VM [33].

The original VM is a homogeneous model. When agents process information, they treat their neighbors equally. From the biological point of view, this is very unreasonable, because biological individuals are inevitably affected by subjective and objective factors when processing information. For example, in the study of the flock of pigeons, it was found that the influence between individuals is not equivalent [35]. In a crowd, individuals prefer to synchronize with people they know. Many studies on non-homogeneous models also show that non-homogeneous models can make the network easier to synchronize [36], [37] and improve the cooperation between individuals [38]. Inspired by this idea, many researchers have proposed weight models from different perspectives. Yang et al. set the weight of neighbors based on the geographical distance of the agents and their neighbors [30]. Gao et al. judge the weight of neighbors by the density of agents around the neighbors [31]. Zou et al. consider that the weight of neighbors should be determined by the angle between the position of neighbor and direction of agents [32].

In this paper, we propose a direction preference model in which agents prefer to synchronize with their neighbors moving in a certain direction. The center agent judges the influence value from its neighbor according to the direction angle between them. We assume that there exists a preference direction angle $\beta$. When the direction angle approaches $\beta$, the neighbor has greater influence on the center agent. In this paper, we concentrate our research on the influence of different $\beta$ on convergence time; whereas, Romensky et al. focus their research on the effect of the velocity restriction on the phase transition of the VM at the case $\beta =0$ [34]. The parameter $\alpha$ $(0<\alpha \le 1)$ is used to adjust the effect of direction preference. The larger the value of $\alpha$, the weaker the effect of direction preference. When $\alpha =1$, the direction preference model will become the original Vicsek model. The effect of the direction preference model on the collective motion is investigated via a large number of simulation experiments under different conditions of restriction angle, absolute velocity, and combination parameter consisting of communication radius and density. Simulation results show that compared to the original VM, in noise-free environment the direction preference model with optimal $\beta =3\pi /8$ can accelerate the synchronization speed; in noise environment the direction preference model with $\beta$ be in $[3\pi /8,5\pi /8]$ has stronger robustness and the optimal value of $\beta$ is altered.